Cedar wood is a good deodorant. I have had good results with using cedar wood shavings inside of a small cloth pouch. Put the pouch inside the jacket while the jacket is laying flat. Zip up the jacket and leave it for a few days (or longer). Hopefully, this will help you.

Miles,
If it is that bad I have seen posts where people have actually machine washed a shearling jacket and air dried. I did some research a while back and bought some lanolin soap which is supposed to be more delicate to shearling. I used it to spot clean cuffs with reasonable results. In a washing machine I imagine greater results.
Dave